Euler's work attracted Lagrange's attention. 1750 and 19-year-old Lagrange became interested in variational methods. He abandoned the geometric analysis and demonstration of Bernoulli brothers and Euler, and introduced the pure analysis method. 1755, he proposed a systematic general method, which he called variational method in his letter to Euler and variational method in the paper submitted to Berlin Academy of Sciences the following year.

For the basic problem of variational method: to maximize or minimize the integral, Lagrange's innovation is to introduce a new curve through the endpoint (x 1, y 1)(x2, y2). He expressed the new curve as Y(X)+Δ y (x), and he introduced the Δ symbol to represent the change of the whole curve y (x), thus obtaining it, which was then expanded by Taylor. Lagrange then proved that, like a univariate function, the maximization function y(x) has δJ=0. He also said that Δ y' = d (Δ y)/dx, that is, the operation order of d and Δ can be interchanged, and then he concluded that the Δ y coefficient must be 0, or Euler once got the equation, which is a necessary and sufficient condition for y(x). In 100 years after Lagrange, the fact that Δ y = 0 was either intuitively accepted or wrongly proved. Even Cauchy's proof is not sufficient. Pierre Frederic Sarus gave the first correct proof in 1848. This result is the basic lemma of variational method.

In 1760/ 176 1, Lagrange first derived the end conditions that the minimization curve with variable end points must meet, and also found the cross-section conditions that the intersection of the minimization curve and the fixed curve or surface must be established. Lagrange's second innovation is the problem related to multiple integrals mentioned in this paper: the integral form is that Z is a function of X and Y, and P and Q are partial derivatives of Z to X and Y respectively, which requires that J should be a maximum or minimum function Z (x, Y). In this kind of multiple integral problem, the most important problem is to find the surface with the smallest area when the boundary is fixed in some way, such as giving two closed curves that are not self-intersecting in space, and then finding the surface with the smallest area surrounded by these two curves (rotating surface). In 1744, Euler solved the special case that the boundary is a circular curve parallel to the yz plane and the center is on the X axis. Lagrange obtained the differential equation that must be satisfied when z(x, y) takes the minimum value by similar method. The partial differential equation obtained after rewriting with a symbol is called gaspard monge equation: Rr+Ss+Tt=U (see/kloc-partial differential equation (VI) in the 8th century). This equation is difficult to solve, and it was a research topic before Euler's time. Lagrange gave the partial differential equation in the minimum surface problem, namely 1785. Mesny pointed out that this partial differential equation shows that at any point on the minimal surface, the radii of principal curvatures are equal and in opposite directions, or the average curvature (the average of principal curvatures) is 0.

In 1770, Lagrange studied the single and multiple integrals with higher derivatives in the integrand. Since Lagrange, this subject has developed into the standard content of variational method, and the principle is the same as above.

Except for Euler and Lagrange, the variational method was not understood by other mathematicians of his time. Euler elaborated Lagrange's method in many works, and used this method to explain several previous results. Although he realized that variational method is a new branch, or a new skill of using δ operator, like Lagrange, he tried to base the logic of variational method on ordinary calculus. He introduced the parameter T and expressed Δ y as a part of WeChat service about T, and got the same result.

1779, Euler continued to study the spatial curve with the property of maximum or minimum. In 1780, he studied the steepest descent line in three-dimensional space under the action of external force (usually gravity) or in the presence of damping medium.